Preface

This is not a proceedings of the 2004 conference “n-Categories: Foundations and Applications” that we organized and ran at the IMA during the two weeks June 7–18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. We also thank the IMA staff for a magnificent job. There has been a great deal of work in higher category theory since then, but we still feel that it is not yet time to offer a volume devoted to the main topic of the conference. At that time, we felt that we were at the beginnings of a large new area of mathematics, but one with many different natural approaches in desperate need of integration into a cohesive field. We feel that way still.

So, instead of an introduction to higher category theory, we have decided to publish a series of papers that provide useful background for this subject. This volume is aimed towards the wider mathematical community, rather than those knowledgeable in category theory and especially higher category theory. We are particularly sensitive to the paucity of young Americans knowledgeable enough in the subject to be potential readers.

The focus of the conference was on comparing the many approaches to higher category theory. These approaches, as they existed at the time of the conference, have been summarized by Leinster (Lein1) and by Cheng and Lauda (ChLau). The earliest was based on filtered simplicial sets. It is due to Street (Str1, Str3) and is being developed in detail by Verity (Verity1, Verity2, Verity3), with a contribution by Gurski (Gur). Another approach, called “opetopic’‘ since operads were used to describe the shapes of diagrams used in the theory, is due to Baez and Dolan (BD) and has been developed by Leinster (Lein2), Cheng (Ch1, Ch2, Ch3, Ch4), and Makkai and his collaborators (HMP, Makkai1). There is also a topologically motivated approach operadsdue to Trimble, which has been studied and generalized by Cheng and Gurski (Ch5, ChGur). There is a quite different and more extensively developed operadic approach to weak ∞-categories due to Batanin (Bat1, Str2), with a variant due to Leinster (Lein3). Penon (Penon) gave a related, very compact definition of infinity-category; this definition was later corrected and improved by Batanin (Bat2) and Cheng and Makkai (ChMakkai). Another highly developed approach, based on n-foldsimplicial sets and inspired by work of Segal in infinite loop space theory, is due to Tamsamani and Simpson (Simpson1, Simpson2, Simpson3, Simpson4, Tam1). Yet another theory, due to Joyal (Berger1, Joyal), is based on a presheaf category called the category of ”theta-sets“. Also, Lurie (Lurie3) has begun making extensive use of Barwick's approach to (∞,n)-categories, which are roughly weak ∞\infty-categories where all morphisms above dimension n are invertible (Barwick).

The theme of the 2004 conference was comparisons among all these approaches to higher categories — or at least, those that existed at the time. While there are papers that tackle aspects of this immense unification project (Ch3, Ch5, Simpson5), it is still quite unclear how a unified theory of higher categories will evolve. In proposing the conference, we wrote as follows: “It is not to be expected that a single all embracing definition that is equally suited for all purposes will emerge. It is not a question as to whether or not a good definition exists. Not one, but many, good definitions already do exist, although they have been worked out to varying degrees. There is growing general agreement on the basic desiderata of a good definition of n-category, but there does not yet exist an axiomatization, and there are grounds for believing that only a partial axiomatization may be in the cards.”

One can make analogies with many other areas where a number of interrelated definitions exist. In algebraic topology, there are various symmetric monoidal categories categories of spectra, and they are related by a web of Quillen equivalences of model categories. In this context, all theories are in some sense “the same”, but the applications require use of different models: many things that can be proven in one model cannot easily be proven in another (May). In algebraic geometry, there are many different cohomology theories, definitely not all the same, but connected by various comparison functors. Motivic theory is in part a search for a universal source of such comparisons.

In the case of weak nn-categories, it is unclear whether there is a useful sense in which all known theories are the same. We do not have a complete web of comparison maps relating different theories. Nor are we sure what it means for two theories to be “the same”, despite important insights by Grothendieck (Gro) and Makkai (Makkai2). The terms in which comparisons should be made are not yet clear. Quillen model category theory should capture some comparisons, but it may be too coarse to give the complete story. A smaller related theme of the conference was that there should be a “baby” comparison project, for which model category theory would in fact be sufficient. Precisely, the idea was that there should be a web of Quillen equivalences among the various notions of (∞,1)-category. These include topologically or simplicially enriched categories, Segal categories, complete Segal spaces, and quasi-categories. In the years since the conference, this comparison project has been largely completed by Bergner (Be1, Be2) and Joyal and Tierney (JT).

Another smaller related theme was the higher categorical modelling ofn-types of topological spaces. It was a dream of Grothendieck (Gro) that weak n-groupoids should model n-types. Brown has shown that strict n-groupoids are easy to compute with (BHS); unfortunately, they capture only part of the information in an n-type. However, Loday (Loday) and others have found other strict algebraic structures that can fully model n-types. More recently we are seeing work that implements Grothendieck’s original idea in various approaches to weak n-categories and that compares these approaches to the algebraic approaches (Bat3, Bat4, Berger2, Cisinski, Pa1, Pa2, Tam2).

As mentioned, the goal of this volume is merely to prepare the reader for more detailed study of these fast-moving topics. So, we begin with a light-hearted paper that treats Grothendieck's dream as a starting-point for speculations on the relationship between nn-categories and cohomology. It is based on notes that Michael Shulman took of John Baez’s 2007 Namboodiri Lectures at Chicago. Higher category theory has largely developed from a series of analogies with and potential applications to other subjects, including algebraic topology, algebraic geometry, mathematical physics, computer science, logic, and, of course, category theory. This paper illustrates this, and raises the challenge of formalizing the patterns that become visible thereby.

Logically, we might next delve into various approaches to nn-categories and full-fledged ∞-categories. But this seems premature. So instead, the rest of the volume goes back to the beginnings of the subject. By now every well-educated young mathematician can be expected to be familiar with categories, as introduced by Eilenberg and Mac Lane in 1945. It has taken longer to understand that what they introduced was a 2-category: Cat, with categories as objects, functors as morphisms, and natural transformations as 2-morphisms. Ehresmann (Ehresmann) introduced strict n-categories sometime in the 1960’s, and Eilenberg and Kelly discussed them in 1965 (EK), with Cat as a key example of a strict 2-category. Bénabou introduced the more general weak 2-categories or “bicategories” the following year (Benabou). But even today, the mathematics of 2-categories is considered somewhat recondite, even by many mathematicians who implicitly use these structures all the time. There is a great deal of basic 2-categories theory that can illuminate everyday mathematics. The second author rediscovered a chunk of this while writing a book on parametrized homotopy theory (MS), and he was chastened to see how little he knew of something that was so very basic to his own work.

For this reason, the next paper is the longest in this volume: a thorough introduction to the theory of 2-categories, by Steve Lack. This paper gives a solid grounding for anyone who wants some idea of what lies beyond mere categories and how to work with higher categorical notions. Anybody interested in higher category theory must learn something of the richness of 2-categories.

Lawrence Breen’s paper, on gerbes and 2-gerbes, gives an idea of how naturally 2-categorical algebra arises in the study of algebraic and differential geometry. His paper also illustrates the need for “enriched” higher category theory, in which one deals with hom objects that have more structure than is seen in merely set-based categories. Many more such applications could be cited.

Steve Lack is an Australian, and it is noteworthy that the premier world center for category theory has long been Sydney. We have dedicated this volume to Max Kelly, the founder of the Australian school of category theory, who died in 2007. Kelly visited Chicago in 1970–71, just before becoming Head of the Department of Pure Mathematics at the University of Sydney. That was long before e-mail, and Max was considering how best to build up a department that would necessarily suffer from a significant degree of isolation. He succeeded admirably. The final paper in this volume, by Kelly’s student Ross Street, gives a fascinating mathematical and personal account of the development of higher category theory in Australia.

We had very much hoped to include a survey by André Joyal of his important work on quasi-categories. As shown in the work of Lurie (Lurie1, Lurie2), these give a very tractable model of (∞,1)-categories. Joyal’s work showing that one “can do category theory” in quasi-categories is an essential precursor to Lurie’s work and is unquestionably one of the most important recent developments in higher category theory. However, Joyal’s survey is not yet complete: it has grown to hefty proportions and is still growing. So, it will appear separately.